Problem: Solve the equation. $\dfrac{dy}{dx}=-10x^4y$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=-2(x+C)^5$ (Choice B) B $y=Ce^{-2x^5}$ (Choice C) C $y=-2x^5+C$ (Choice D) D $y=e^{-2x^5}+C$
We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=-10x^4y \\\\ \dfrac{1}{y}\,dy&=-10x^4\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} \dfrac{1}{y}\,dy&=-10x^4\,dx \\\\ \int \dfrac{1}{y}\,dy&=\int -10x^4\,dx \\\\ \ln|y|&=-2x^5+C_1 \\\\ e^{\ln|y|}&=e^{-2x^5+C_1} \\\\ |y|&=e^{-2x^5}e^{C_1} \\\\ y&=Ce^{-2x^5} \end{aligned}$ [Where did we get C?] Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=Ce^{-2x^5}$